Prove or disprove the following statement using sets frontier points

[ppy]

if A is a subset of B and the frontier of B is a subset of A then A=B.

I am pretty sure that this is true as I drew I diagram and I think this helped.

A frontier point has a sequence in the set and a sequence in the compliment that both converge to the same limit. However I'm not really sure how to use this definition to help me

Thanks

 

[R136a1]

What if A is the frontier of B?

 

[economicsnerd]

So you have $\partial B \subseteq A\subseteq B$. In particular, $\partial B \subseteq B$ exactly says that $B$ is closed. So $B$ is a closed set, and $A$ is a subset of $B$ which includes every non-interior point of $B$.

The case R136a1 mentioned is, in some sense, the most extreme possible case of $A\neq B$ (a counterexample to your conjecture, as long as $B$ has nonempty interior).

 

[Robert1986]

As a side question, what the heck is a "frontier" of a set? This looks equivalent to the definition of boundary. Is this just another word for boundary? If so, why?

That is, why have a new word?

 

[HallsofIvy]

As both R136a1 and economicsnerd said, you can't prove it. Without some qualification, it is not true:
Counterexample: Let A= {0, 1}, B= [0, 1].
(I am assuming that "frontier" is the same as "boundary".)